DAVID A. BELSLEY
Boslon College
EDWIN KUH
Massachusetts Institute of Technology
ROY E. WELSCH
Massachusetts Institute of Technology

Contents
1 Introduction and Overview
2 Detecting influential Observations and Outliers
2.1 Theoretical Foundations, 9
Deletion, 12
Single-Row Effects, 12
Coefficients and Fitted Values. The Hat Matrix.
Residuals. Covariance Matrix.
Differentiation, 24
A Geometric View, 26
Criteria for Influential Observations, 27
Partial-Regression Leverage Plots, 30
Deletion, 31
Studentized Residuals and Dummy Variables, 33
Differentiation, 35
Geometric Approaches, 37
Final Comments, 38
External Scaling. Internal Scaling. Gaps.
Multiple-Row Effects, 31
2.2 Application: an Intercountry Life-Cycle Savings Function,
39
A Diagnostic Analysis of the Model, 39
The Model and Regression Results, 40
Single-Row Diagnostics, 42
Residuals. Leverage and Hat-Matrix Diagonals.
Coefficient Sensitivity. Covariance Matrix Sensitiuity.
Change in Fit. Internal Scaling. A Provisional
Summary.
Xii CONTENTS
Multiple-Row Diagnostics, 5 1
Partial-Regression Leverage Plots: a Preliminary
Analysis. Using Multiple-Row Methodr. Deletion.
Residuals. Differentiation. Geometry.
Final Comments, 63
Appendix 2A: Additional Theoretical Background, 64
Deletion Formulas, 64
Differentiation Formulas, 65
Theorems Related to the Hat Matrix, 66
Size of the Diagonal Elements. Distribution
Theory. Dummy Variables and Singular
Matrices.
Appendix 2B: Computational Elements, 69
Computational Elements for Single-Row
Diagnostics, 69
Orthogonal Decompositions, the Least-Squares
Solution, and Related Statistics. The Diagonal
Elements of the Hat Matrix. Computing the
DFBETA.
Computational Elements for Multiple-Row
Diagnostics, 75
Notation and the Subset Tree. An Algorithm
for the Geometric Measure, Wilk’ A Dummy
Variables, Sequential Choleski Decomposition,
and the Andrews-Pregibon Statistic. Further
Elements Computed from the Triangular
Factors. Inequalities Related to MDFFIT.
3 Detecting and Assessi Collinearity 85
3.1 Introduction and Historical Perspective, 85
Overview, 91
Historical Perspective, 92
A Basis for a Diagnostic, 96
The Singular-Value Decomposition, 98
3.2 Technical Background, 98
Exact Linear Dependencies: Rank Deficiency, 99
The Condition Number, 100
Near Linear Dependencies: How Small is Small?, 104
The Regression-Coefficient Variance Decomposition, 105
Two Interpretive Considerations, 107
CONTENTS xiii
Near Collinearity Nullified by Near Orthogonality, 107
At Least Two Variates Must Be Involved, 108
An Example, 110
The Diagnostic Procedure, 112
Examining the Near Dependencies, 113
What is “Large” or “High,” 114
The I11 Effects of Collinearity, 114
Computational Problems, 1 14
Statistical Problems, 115
Harmful Versus Degrading Collinearity, 1 15
3.3 Experimental Experience, 1 17
A Suggested Diagnostic Procedure, 112
The Experimental Procedure, 117
The Choice of the X’s, 119
Experimental Shortcomings, 1 19
The Need for Column Scaling, 120
The Experimental Report, 121
The Individual Experiments, 121
The Results, 125
3.4 Summary Interpretation, and Examples of Diagnosing
Actual Data for Collinearity, 152
Interpreting the Diagnostic Results: a Summary of the
Experimental Evidence, 152
Experience with a Single Near Dependency, 153
Experience with Coexisting Near Dependencies, 154
Employing the Diagnostic Procedure, 156
The Steps, 157
Forming the Auxiliary Regressions, 159
Software, 160
Applications with Actual Data, 160
The Bauer Matrix, 161
The Consumption Function, 163
The Friedman Data, 167
An Equation of the IBM Econometric Model, 169
The Condition Number and Invertibility, 173
Parameterization and Scaling, 177
The Effects on the Collinearity Diagnostics
Due to Linear Transformations of the Data,
177
Appendix 3A:
Appendix 3B:
XiV CONTENTS
Each Parameterization is a Different Problem,
178
A More General Analysis, 180
Column Scaling, 183
Appendix 3C: The Weakness of Correlation Measures in
Providing Diagnostic Information, 185
Appendix 3D: The Harm Caused by Collinearity, 186
The Basic Harm, 187
The Effect of Collinearity, 190
4 Applications and Remedies 192
4.1 A Remedy for Collinearity: the Consumption Function
with Mixed-Estimation, 193
Corrective Measures, 193
Introduction of New Data, 193
Bayesian- Type Techniques, 194
Pure Bayes. Mixed-Estimation. Ridge Regression.
Application to the Consumption-Function Data, 196
Prior Restrictions, 197
Ignored Information, 199
Summary of Prior Data, 200
Regression Results and Variance-Decomposition
Proportions for Mixed-Estimation Consumption-Function
Data, 200
4.2 Row-Deletion Diagnostics with Mixed-Estimation of the
U.S. Consumption Function, 204
A Diagnostic Analysis of the Consumption-Function
Data, 204
Single-Row Diagnostics, 205
Residuals. Leverage and Hat-Matrix Diagonals.
Coefficient Sensitiuity.
Summary, 207
A Reanalysis after Remedial Action for Ill Conditioning,
207
The Row Diagnostics, 208
A Suggested Research Strategy, 210
4.3 An Analysis of an Equation Describing the Household
Demand for Corporate Bonds, 212
CONTENTS xv
An Examination of Parameter Instability and Sensitivity,
215
Tests for Overall Structural Instability, 215
Sensitivity Diagnostics, 2 17
The Monetary Background, 219
A Use of Ridge Regression, 219
Summary, 228
Residuals. Leoerage and Coefficient Sensitivity.
4.4 Robust Estimation of a Hedonic Housing-Price Equation,
229
The Model, 231
Robust Estimation, 232
Partial Plots, 235
Single-Row Diagnostics, 237
Multiple-Row Diagnostics, 241
Summary, 243
Appendix 4A: Harrison and Rubinfeld Housing-Price Data,
245
5 Research Issues and Directions for Extensions
5.1 Issues in Research Strategy, 263
5.2 Extensions of the Diagnostics, 266
Extensions to Systems of Simultaneous Equations, 266
Influential-Data Diagnostics, 266
Collinearity Diagnostics, 268
Extensions to Nonlinear Models, 269
Influential-Data Diagnostics, 269
Collinearity Diagnostics, 272
Bounded-Influence Regression, 274
Multiple-Row Procedures, 274
Transformations, 275
Time Series and Lags, 276
Additional Topics, 274
262
Bibliography
Author Index
Subject Index